Flat semilattices

Abstract

Let $A$, $B$, and $S$ be (v,0)-semilattices and let $f:A\to B$ be a (v,0)-embedding. Then the canonical map, $f \otimes \id\_S$, of the tensor product $A\otimes S$ into the tensor product $B\otimes S$ is not necessarily an embedding. The (v,0)-semilattice $S$ is flat, if for every embedding $f:A\to B$, the canonical map $f\otimes\id$ is an embedding. We prove that a (v,0)-semilattice $S$ is flat if and only if it is distributive.